(* Title: HOL/Library/Option_ord.thy
ID: $Id$
Author: Florian Haftmann, TU Muenchen
*)
header {* Canonical order on option type *}
theory Option_ord
imports Plain
begin
instantiation option :: (order) order
begin
definition less_eq_option where
[code del]: "x \<le> y \<longleftrightarrow> (case x of None \<Rightarrow> True | Some x \<Rightarrow> (case y of None \<Rightarrow> False | Some y \<Rightarrow> x \<le> y))"
definition less_option where
[code del]: "x < y \<longleftrightarrow> (case y of None \<Rightarrow> False | Some y \<Rightarrow> (case x of None \<Rightarrow> True | Some x \<Rightarrow> x < y))"
lemma less_eq_option_None [simp]: "None \<le> x"
by (simp add: less_eq_option_def)
lemma less_eq_option_None_code [code]: "None \<le> x \<longleftrightarrow> True"
by simp
lemma less_eq_option_None_is_None: "x \<le> None \<Longrightarrow> x = None"
by (cases x) (simp_all add: less_eq_option_def)
lemma less_eq_option_Some_None [simp, code]: "Some x \<le> None \<longleftrightarrow> False"
by (simp add: less_eq_option_def)
lemma less_eq_option_Some [simp, code]: "Some x \<le> Some y \<longleftrightarrow> x \<le> y"
by (simp add: less_eq_option_def)
lemma less_option_None [simp, code]: "x < None \<longleftrightarrow> False"
by (simp add: less_option_def)
lemma less_option_None_is_Some: "None < x \<Longrightarrow> \<exists>z. x = Some z"
by (cases x) (simp_all add: less_option_def)
lemma less_option_None_Some [simp]: "None < Some x"
by (simp add: less_option_def)
lemma less_option_None_Some_code [code]: "None < Some x \<longleftrightarrow> True"
by simp
lemma less_option_Some [simp, code]: "Some x < Some y \<longleftrightarrow> x < y"
by (simp add: less_option_def)
instance by default
(auto simp add: less_eq_option_def less_option_def split: option.splits)
end
instance option :: (linorder) linorder
by default (auto simp add: less_eq_option_def less_option_def split: option.splits)
end